By Theorem 3.2 and 4.1 of [1]},
\( \mathrm {corank}_{\Lambda (G)}(H^1(G_{\infty }^{S}, E_{p^{\infty }}))=[L:\mathbb {Q}]=2. \) 
Putting this together with Lemma REF -[lem:mu-inv-H1-2](2),
\(\mathrm {corank}_{\Lambda (G)}(H^1(G_{\infty }^{S}, E_{p^{\infty }}))=\mathrm {corank}_{\Omega (G)}(H^1(G_{\infty }^{S}, E_{p}))=2.\) 

For a prime \( w \nmid p \)  in \( L_\infty \)  Proposition 2 in [2]} implies that
\( \Lambda (G) \) -corank of \( H^{1}(L_{\infty ,w},E_{p^{\infty }}) \)  is zero.
This means that for \( v \nmid p \) , the \( \Lambda (G) \) -corank of \( J_v^{\pm }(E_{p^{\infty }}/L_\infty ) \)  is zero.
Hence,
\(\mathrm {corank}_{\Lambda (G)}(\displaystyle \bigoplus _{v \in S} J_v^{\pm }(E_{p^{\infty }}/L_\infty ))=\mathrm {corank}_{\Lambda (G)}(\displaystyle \bigoplus _{v \in S_p} J_v^{\pm }(E_{p^{\infty }}/L_\infty )).\) 

Furthermore, by Lemma REF -[lem:mu-inv-H1-2](2)
\(\mathrm {corank}_{\Lambda (G)}( H^{1}(L_{\infty ,w},E_{p^{\infty }}))&=\mathrm {corank}_{\Omega (G)}H^{1}(L_{\infty ,w}, E_{p})\\=\mathrm {corank}_{\Omega (G)}( ^{\pm }\widetilde{K}_v(E_{p}/L_\infty ))&=0\\\Rightarrow \mathrm {corank}_{\Omega (G)}(\displaystyle \bigoplus _{v \in S} {}^{\pm }\widetilde{K}_v(E_{p}/L_\infty ))&=\mathrm {corank}_{\Omega (G)}(\displaystyle \bigoplus _{v \in S_p} {}^{\pm }\widetilde{K}_v(E_{p}/L_\infty ))\) 

For \( v | p \) , recall the map \( \varphi _{v}^{\pm } \)  (cf. (REF )) gives an isomorphism
between \( (J_v^{\pm }(E_{p^{\infty }}/L_\infty ))_p \)  and \( {^{\pm }\widetilde{K}_v(E_{p}/L_\infty )} \) .
Therefore,
\( \Omega (G) \) -corank of \((J_v^{\pm }(E_{p^{\infty }}/L_\infty ))_p\) 
is equal to
\(\mathrm {corank}_{\Omega (G)}( {^{\pm }\widetilde{K}_v(E_{p}/L_\infty )} ). \) 
Moreover, for \( v | p \)  the local condition \( J_v^{\pm }(E_{p^{\infty }}/L_\infty ) \)  is a free \( \Lambda (G) \) -module (cf. [3]}) and
\(\mathrm {corank}_{\Lambda (G)}(\displaystyle \bigoplus _{v \in S_p} J_v^{\pm }(E_{p^{\infty }}/L_\infty ))=\mathrm {corank}_{\Lambda (G)}(H^1(G_{\infty }^{S}, E_{p^{\infty }})=2.\) 
This means that \( J_v^{\pm }(E_{p^{\infty }}/L_\infty ) \)  has \( \mu \) -invariant zero.
By equation (REF ),
\(\mathrm {corank}_{\Lambda (G)}(J_v^{\pm }(E_{p^{\infty }}/L_\infty ))&=\mathrm {corank}_{\Omega (G)}((J_v^{\pm }(E_{p^{\infty }}/L_\infty ))_p)\\&=\mathrm {corank}_{\Omega (G)}( {^{\pm }\widetilde{K}_v(E_{p}/L_\infty )} ).\) 

This, together with equation (REF ) gives us that
\(\mathrm {corank}_{\Omega (G)}(\displaystyle \bigoplus _{v \in S} {}^{\pm }\widetilde{K}_v(E_{p}/L_\infty ))=\mathrm {corank}_{\Omega (G)}(\displaystyle \bigoplus _{v \in S_p} {}^{\pm }\widetilde{K}_v(E_{p}/L_\infty ))=2.\) 

Comparing the above equation and equation (REF ) yields
\(\mathrm {corank}_{\Omega (G)}(H^1(G_{\infty }^{S}, E_{p})) = \mathrm {corank}_{\Omega (G)}(\displaystyle \bigoplus _{v \in S} {^{\pm }\widetilde{K}_v(E_{p}/L_\infty )})=2.\) 

The exact sequence (REF ) implies that
\(\mathrm {corank}_{\Omega (G)}(\mathcal {R}^{\pm /\pm } (E_p/L_\infty )) = \mathrm {corank}_{\Omega (G)}({R}^{\pm /\pm }(E_p/ L_{\infty })^{\wedge }).\) 
